When you roll a fair 6 sided die, the expected number of rolls to get a certain number (say 6) is 6. But the memoryless property says that if you roll any amount of non-6 numbers then you'd still expect to have 6 more rolls until you see 6.
What if you wanted to bet on how many rolls it would take to get 6 and you were allowed to change your guess after each roll (as long as it hasn't turned up 6 yet and the number of rolls hasn't reached your guess). So you'd guess 6 rolls at the start, and the dice comes up a 4. Do you update your guess to 7 rolls total or keep the initial guess? The memoryless says we should update but if you kept changing your guess to be 6 rolls from the current roll you'd never win! What would be the optimal strategy here?
Based on your clarification of the rules in the comments, if you guess more than $1$, and always update on a miss by adding more than $1$, then as you noted, you will never win.
It follows that in order to win, at some point, you must wait it out.
Based on that, it's clear that no strategy can yield a probability of winning more than $1/6$.
If your initial guess is $1$, then you win with probability $1/6$.
If your initial guess is more than $1$, then if a $6$ appears on the first roll, you lose, hence no strategy with an initial guess of more than $1$ has a winning probability more than $(5/6){\,\cdot\,}(1/6)=5/36$.
It follows that the unique optimal strategy is to make an initial guess of $1$ (and then you win or lose based on the result of the first roll).